We consider boundary value problems for the N -wave interaction equations in one and two space dimensions, posed for x ≥ 0 and x, y ≥ 0 respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the bou...

Many important problems in the theory of integrable systems and approximation theory can be recast as Riemann-Hilbert problems for a matrix-valued unknown. Via the connection with approximation theory, and specifically the theory of orthogonal polynomials, one can also study problems from the theory of random matrix ensembles and combinatorics. Roughly speaking, solving a Riemann-Hilbert proble...

We establish the existence of a real solution y(x, T ) with no poles on the real line of the following fourth order analogue of the Painlevé I equation, x = Ty − ( 1 6 y + 1 24 (y x + 2yyxx) + 1 240 yxxxx ) . This proves the existence part of a conjecture posed by Dubrovin. We obtain our result by proving the solvability of an associated Riemann-Hilbert problem through the approach of a vanishi...

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express th...